The Function PT and QT Are Continuous for Every T It is Stated That Et 1 and LnT1
The Student t distribution is one of the most commonly used distribution in statistics. This tutorial explains how to work with the Student t distribution in R using the functions dt(),qt(),pt(), andrt().
dt
The function dtreturns the value of the probability density function (pdf) of the Student t distribution given a certain random variable xand degrees of freedom df. The syntax for using dt is as follows:
dt(x, df)
The following code illustrates a few examples of dtin action:
#find the value of the Student t distribution pdf at x = 0 with 20 degrees of freedom dt(x = 0, df = 20) #[1] 0.3939886 #by default, R assumes the first argument is x and the second argument is df dt(0, 20) #[1] 0.3939886 #find the value of the Student t distribution pdf at x = 1 with 30 degrees of freedom dt(1, 30) #[1] 0.2379933
Typically when you're trying to solve questions about probability using the Student t distribution, you'll often use ptinstead of dt. One useful application of dt, however, is in creating a Student t distribution plot in R. The following code illustrates how to do so:
#Create a sequence of 100 equally spaced numbers between -4 and 4 x <- seq(-4, 4, length=100) #create a vector of values that shows the height of the probability distribution #for each value in x, using 20 degrees of freedom y <- dt(x = x, df = 20) #plot x and y as a scatterplot with connected lines (type = "l") and add #an x-axis with custom labels plot(x,y, type = "l", lwd = 2, axes = FALSE, xlab = "", ylab = "") axis(1, at = -3:3, labels = c("-3s", "-2s", "-1s", "mean", "1s", "2s", "3s"))
This generates the following plot:
pt
The function ptreturns the value of the cumulative density function (cdf) of the Student t distribution given a certain random variable xand degrees of freedomdf. The syntax for using pnorm is as follows:
pt(x, df)
Put simply, ptreturns the area to the left of a given valuexin the Student t distribution. If you're interested in the area to the right of a given valuex, you can simply add the argument lower.tail = FALSE
pt(x, df, lower.tail = FALSE)
The following examples illustrates how to solve some probability questions using pt.
Example 1: Find the area to the left of a t-statistic with value of -0.785 and 14 degrees of freedom.
pt(-0.785, 14) #[1] 0.2227675 Example 2: Find the area to the right of a t-statistic with value of -0.785 and 14 degrees of freedom.
#the following approaches produce equivalent results #1 - area to the left 1 - pt(-0.785, 14) #[1] 0.7772325 #area to the right pt(-0.785, 14, lower.tail = FALSE) #[1] 0.7772325 Example 3: Find the total area in a Student t distribution with 14 degrees of freedom that lies to the left of -0.785 or to the right of 0.785.
pt (-0.785, 14) + pt (0.785, 14, lower.tail = FALSE) #[1] 0.4455351
qt
The function qtreturns the value of the inverse cumulative density function (cdf) of the Student t distribution given a certain random variable xand degrees of freedomdf.The syntax for using qt is as follows:
qt(x, df)
Put simply, you can use qtto find out what the t-score is of the pth quantile of the Student t distribution.
The following code illustrates a few examples of qtin action:
#find the t-score of the 99th quantile of the Student t distribution with df = 20 qt(.99, df = 20) # [1] [1] 2.527977 #find the t-score of the 95th quantile of the Student t distribution with df = 20 qt(.95, df = 20) # [1] 1.724718 #find the t-score of the 90th quantile of the Student t distribution with df = 20 qt(.9, df = 20) # [1] 1.325341
Note that the critical values found byqtwill match the critical values found in the t-Distribution table as well as the critical values that can be found by the Inverse t-Distribution Calculator.
rt
The function rtgenerates a vector of random variables that follow a Student t distribution given a vector lengthnand degrees of freedomdf. The syntax for using rt is as follows:
rt(n, df)
The following code illustrates a few examples of rtin action:
#generate a vector of 5 random variables that follow a Student t distribution #with df = 20 rt(n = 5, df = 20) #[1] -1.7422445 0.9560782 0.6635823 1.2122289 -0.7052825 #generate a vector of 1000 random variables that follow a Student t distribution #with df = 40 narrowDistribution <- rt(1000, 40) #generate a vector of 1000 random variables that follow a Student t distribution #with df = 5 wideDistribution <- rt(1000, 5) #generate two histograms to view these two distributions side by side, and specify #50 bars in histogram, par(mfrow=c(1, 2)) #one row, two columns hist(narrowDistribution, breaks=50, xlim = c(-6, 4)) hist(wideDistribution, breaks=50, xlim = c(-6, 4))
This generates the following histograms:
Notice how the wide distribution is more spread out compared to the narrow distribution. This is because we specified the degrees of freedom in the wide distribution to be 5 compared to 40 in the narrow distribution. The fewer degrees of freedom, the wider the Student t distribution will be.
Further Reading:
A Guide to dnorm, pnorm, qnorm, and rnorm in R
A Guide to dbinom, pbinom, qbinom, and rbinom in R
Source: https://www.statology.org/working-with-the-student-t-distribution-in-r-dt-qt-pt-rt/
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